"....using a Blackberry, everything is routed via what is essentially Blackberry's server. This seems illogical to me, so I'm obviously missing something."
BlackBerry became famous by providing a fast and secure 'push' email service via a BlackBerry mobile phone for business users. The email service hasn't changed, but it's available to anyone who buys a BlackBerry, rather than just business users.
Once you've set up your email account information on your BlackBerry it periodically looks at your POP mailbox (every 15 minutes by default), and if it finds inbound messages it 'pushes' them to your phone via the BlackBerry mail servers. These servers are secure, and trusted by millions of BlackBerry users - or at least they were, until recently.
What has happened over the past week or so is that the BlackBerry mail servers have crashed, or at least some of them have. Blackberry runs separate servers for business and personal users, and it's the personal servers that have been at fault. There's been a good deal of understandable dissatisfaction amongst BlackBerry users (I'm one myself), but it will probably soon blow over. BlackBerry phones are so good that there's a considerable amount of consumer loyalty, and this should stand RIM (the BlackBerry parent company) in good stead at an awkward time. The company has been struggling to conquer the home-user market for some time.